Inverse spectral results on two-dimensional tori

is much more complicated than the corresponding problem on surfaces of genus > 2. On one hand, for certain classes of potentials, one can reconstruct V from the spectrum of (1.1); on the other hand, there are potentials that have (lots of) isospectral deformations. In their analysis of (1. 1), an important element is the decomposition of the Fourier series of V into "primitive summands," and we begin here by saying a few words about this decomposition since it will also be important in the results to be described. An element of a lattice is said to be primitive if it is not a positive integer multiple of another element of the lattice. Now let a e27i(w,x) Ct E r,